Question

the annual profits for a company are given in the following table, where x represents the number of years since 2002, and y represents the profit in thousands of dollars. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. using this equation, find the projected profit (in thousands of dollars) for 2010, rounded to the nearest thousand dollars.

years since 2002 (x)	profit (y) (in thousands of dollars)
1	160
2	157
3	192
4	192
5	227

Explanation:

Step1: Calculate sums

Let \(n = 6\) (number of data - points).
\(\sum_{i = 1}^{n}x_{i}=0 + 1+2 + 3+4 + 5=\frac{5\times(5 + 1)}{2}=15\)
\(\sum_{i = 1}^{n}y_{i}=135 + 160+157+192+192+227 = 1063\)
\(\sum_{i = 1}^{n}x_{i}^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=\frac{5\times(5 + 1)\times(2\times5 + 1)}{6}=55\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=0\times135+1\times160 + 2\times157+3\times192+4\times192+5\times227=0 + 160+314+576+768+1135 = 2953\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the linear - regression line \(y=mx + b\) is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\)
Substitute the values:
\[

$$\begin{align*} m&=\frac{6\times2953-15\times1063}{6\times55 - 15^{2}}\\ &=\frac{17718-15945}{330 - 225}\\ &=\frac{1773}{105}\\ &\approx16.89 \end{align*}$$

\]

Step3: Calculate y - intercept \(b\)

The formula for the y - intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}\)
Substitute \(m\approx16.89\), \(\sum_{i = 1}^{n}x_{i}=15\), \(\sum_{i = 1}^{n}y_{i}=1063\) and \(n = 6\)
\[

$$\begin{align*} b&=\frac{1063-16.89\times15}{6}\\ &=\frac{1063 - 253.35}{6}\\ &=\frac{809.65}{6}\\ &\approx134.94 \end{align*}$$

\]

The linear - regression equation is \(y = 16.89x+134.94\)

Step4: Find \(x\) value for 2010

Since \(x\) represents the number of years since 2002, for 2010, \(x = 2010 - 2002=8\)

Step5: Calculate projected profit

Substitute \(x = 8\) into the equation \(y = 16.89x+134.94\)
\[

$$\begin{align*} y&=16.89\times8+134.94\\ &=135.12+134.94\\ &=270.06\approx270 \end{align*}$$

\]

Answer:

The linear - regression equation is \(y = 16.89x + 134.94\) and the projected profit for 2010 is 270 thousand dollars.